[Paper Review] Conformal geometry of the supercotangent and spinor bundles

Jean-Philippe Michel|arXiv (Cornell University)|Apr 9, 2010

Homotopy and Cohomology in Algebraic Topology49 references3 citations

TL;DR

This paper establishes a geometric quantization framework linking the classical supercotangent bundle and the quantum spinor bundle on conformally flat pseudo-Riemannian manifolds. By constructing a Hamiltonian lift of local conformal vector fields to the supercotangent bundle, it shows that geometric quantization yields the spinor bundle as the quantum representation space, and identifies Kosmann's Lie derivative of spinors as the quantized action. The key contribution is the construction of conf(M,g)-module structures on symbols and differential operators, with conformally equivariant quantization providing an isomorphism between symbol and operator modules.

ABSTRACT

We establish, via geometric quantization of the supercotangent bundle sM of (M,g), a correspondence between its conformal geometry and those of the spinor bundle. In particular, the Kosmann Lie derivative of spinors is obtained by quantization of the comoment map, associated to the new Hamiltonian action of conf(M,g) on sM. We study then the conf(M,g)-module structures induced on the space of differential operators acting on spinor densities and on its spaces of symbols (functions on sM). In the conformally flat case, we classify their conformal invariants, including the conformally odd powers of the Dirac operator.

Motivation & Objective

• To establish a geometric quantization framework for the supercotangent bundle of a conformally flat pseudo-Riemannian manifold (M,g).
• To show that geometric quantization of the supercotangent bundle produces the spinor bundle as the quantum representation space.
• To define a Hamiltonian lift of local conformal vector fields to the supercotangent bundle and identify its quantization as Kosmann's Lie derivative of spinors.
• To construct conf(M,g)-module structures on the spaces of symbols and differential operators acting on spinor densities.
• To generalize conformally equivariant quantization to the spinor setting, establishing an isomorphism between symbol and operator modules preserving principal symbols.

Proposed method

• Utilize the canonical symplectic form on the supercotangent bundle M = T*M ×M ΠTM, constructed via Rothstein's representation theorem for even symplectic supermanifolds.
• Apply geometric quantization to the symplectic supermanifold (M, ω), using a polarization to construct the quantum Hilbert space as the spinor bundle.
• Define the Hamiltonian lift of conformal vector fields to M by lifting the action to the supercotangent bundle using the comoment map.
• Identify the quantum action of conf(M,g) on spinors as the Kosmann Lie derivative, derived from the quantization of the comoment map.
• Construct conf(M,g)-module structures on the filtered spaces of symbols Sδ[ξ], tensorial symbols Tδ[ξ], and differential operators Dλ,µ.
• Prove the existence and uniqueness of a conformally equivariant quantization Qλ,µ: Sδ[ξ] → Dλ,µ for generic weights λ, µ, preserving the principal symbol.

Experimental results

Research questions

• RQ1 How can the classical supercotangent bundle of a pseudo-Riemannian manifold be equipped with a canonical symplectic structure suitable for geometric quantization?
• RQ2 What is the Hamiltonian lift of local conformal vector fields to the supercotangent bundle, and how does it relate to the Lie derivative of spinors?
• RQ3 Can geometric quantization of the supercotangent bundle reproduce the standard construction of the spinor bundle and its covariant derivative?
• RQ4 What is the structure of the conf(M,g)-module of symbols and differential operators acting on spinor densities, and how do they relate via conformally equivariant quantization?
• RQ5 Which conformally invariant differential operators arise from the associated graded module in the conformally flat case, and how do they relate to the GJMS operators and Q-curvature?

Key findings

• Geometric quantization of the supercotangent bundle (M, ω) yields the spinor bundle as the quantum representation space, establishing a canonical link between classical and quantum phase spaces.
• The Kosmann Lie derivative of spinors arises naturally as the quantization of the comoment map, providing a geometric realization of this non-canonical but geometrically natural construction.
• The space of differential operators acting on spinor densities and the space of their symbols both carry conf(M,g)-module structures, with a common associated graded module isomorphic to Tδ[ξ].
• In the conformally flat case, conformally equivariant quantization exists and is unique for generic weights λ, µ, providing an isomorphism between symbol and operator modules preserving the principal symbol.
• The classification of conformal invariants reveals two new conformally odd powers of the Dirac operator, which do not have counterparts in the scalar case and depend on the orientation of M.
• The invariant R ∈ T²ₙ[ξ] has no lift to S²ₙ[ξ], implying the non-existence of a conformally equivariant superization S²ₙᵀ, indicating a fundamental obstruction in the spinor setting.